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A general inexact iterative method for monotone operators, equilibrium problems and fıxed point problems of semigroups in Hilbert spaces

Research paper by Vittorio Colao, Giuseppe Marino, Daya Ram Sahu

Indexed on: 15 May '12Published on: 15 May '12Published in: Fixed Point Theory and Applications



Abstract

Let H be a real Hilbert space. Consider on H a nonexpansive family T={T(t):0≤t<∞}Open image in new window with a common fixed point, a contraction f with the coefficient 0 < α < 1, and a strongly positive linear bounded self-adjoint operator A with the coefficient γ̄>0Open image in new window. Assume that 0<γ<γ̄/αOpen image in new window and that S={St:0≤t<∞}Open image in new window is a family of nonexpansive self-mappings on H such that F(T)⊆F(S)Open image in new window and Open image in new window has property (A)Open image in new window with respect to the family Open image in new window. It is proved that the following schemes (one implicit and one inexact explicit):andconverge strongly to a common fixed point x*∈F(T)Open image in new window, where F(T)Open image in new window denotes the set of common fixed point of the nonexpansive semigroup. The point x* solves the variational in-equality 〈(γf −A)x*, x−x*〉 ≤ 0 for all x∈F(T)Open image in new window. Various applications to zeros of monotone operators, solutions of equilibrium problems, common fixed point problems of nonexpansive semigroup are also presented. The results presented in this article mainly improve the corresponding ones announced by many others.2010 Mathematics Subject Classification: 47H09; 47J25.